Stoquastic Hamiltonians, a Leisurely Introduction The motivation for stoquastic Hamiltonians comes from the success of quantum monte carlo methods. There are many variants of quantum monte carlo but all of them have a similar character, which I will outline here. 1 / 48
We often want to evaluate expressions like this: Tr [ Oe − β H ] Tr [ e − β H ] But accessing e − β H can be nasty. We can break up the exponentials into small steps Tr [ Oe − β H ] = Tr [ O ( e − β L H ) L ] and insert identities. � �� � �� � �� � � e − β e − β L H L H = Tr O | x 1 �� x 1 | | x 2 �� x 2 | | x 3 �� x 3 | .. x 1 x 2 x 3 2 / 48
� �� � �� � �� � � e − β e − β L H L H = Tr | x 1 �� x 1 | | x 2 �� x 2 | | x 3 �� x 3 | O .. x 1 x 2 x 3 If L is large enough, and H is efficiently expressible, then e − β L H = ( I − β L H ) = G is easier to handle. We can factor out the sums, to transform the expression into a sum over “paths” � x = ( x 0 , x 1 , x 2 .. ) L − 1 � � Tr [ Oe − β H ] = � x 0 | O | x 1 � � x i | G | x i +1 � x | x 0 = x L � i =1 Tr [ Oe − β H ] = � O ( x ) w ( x ) x We don’t want to sum over these paths 3 / 48
� Tr [ Oe − β H ] = O ( x ) w ( x ) x The strategy of quantum Monte Carlo is to sample from these paths in a way that is faithful to this weighting. L − 1 Tr [ Oe − β H ] = � � � x 0 | O | x 1 � � x i | G | x i +1 � i =1 x | x 0 = x L � For example [Sorella, Capriotti (2013)], one might have a walker ( w , x ), and one might start at a random x and perform a random walk informed by the matrix elements � x | G | y � = G xy : | G xy | x → y w/ prob. � y | G xy | �� � w → w ∗ sign ( G xy ) ∗ | G xy | y 4 / 48
Sampling in this way will reproduce the distribution, but there is a problem. � x O ( x ) w ( x ) � O � = � x w ( x ) � x O ( x ) sign ( w ( x )) | w ( x ) | �� x | w ( x ) | � � O � = � x sign ( w ( x )) | w ( x ) | � x | w ( x ) | | w ( x ) | Define: P ( x ) := x | w ( x ) | , δ ( x ) := sign ( w ( x )) � � x O ( x ) δ ( x ) P ( x ) = � O δ � � O � = � x δ ( x ) P ( x ) � δ � If there are negative signs in G , then for long path lengths the average sign will tend to zero, and relative errors can blow up. 5 / 48
How do we avoid this “sign problem”? If H is real and has non-positive entries in its off-diagonals then for some sufficiently large L , G = I − β L H is entrywise non-negative and real, so all path weights are positive and real. � δ � = 1 We call such H globally stoquastic (in the standard basis). In fact, e − β H is an entrywise non-negative matrix for all β if and only if H is globally stoquastic. — If H is stoquastic, G is non-negative for large L , therefore e − β H = G L is non-negative. — If e − β H is non-negative for all β , then choose sufficiently small e − β H = I − β H + O ( β 2 || H || 2 ) and so H must have β : non-positive off-diagonals. 6 / 48
If e − β H is positive and real, then the Perron-Frobenius theorem tells us that the ground state of H is a vector with all positive and real weights. � | ψ ( x ) | 2 � x | H | x � + � ψ ( x ) ψ ∗ ( y ) � x | H | y � � ψ | H | ψ � = x x � = y � | ψ ( x ) | 2 � x | H | x � − � ψ ( x ) ψ ∗ ( y ) |� x | H | y �| � ψ | H | ψ � = x x � = y � | ψ ( x ) | 2 � x | H | x � − � | ψ ( x ) || ψ ∗ ( y ) ||� x | H | y �| �| ψ || H || ψ |� = x x � = y �| ψ || H || ψ |� ≤ � ψ | H | ψ � (Thanks to Alex for that proof) We can think of the ground state as the stationary probability distribution of a quantum monte carlo process 7 / 48
Some comments on terminology: ◮ Globally stoquastic in standard basis: � x | H | y � ≤ 0 x � = y ◮ (termwise) Stoquastic in standard basis: H = � : � x | H k | y � ≤ 0 x � = y k H k ◮ Globally stoquastic � = termwise stoquastic in general ◮ For 2-local multi-qubit Hamiltonians they are the same. (but not for 3-local) ◮ Computer scientists seem to care about termwise stoquastic ◮ Monte Carlo community it is not so clear to me. Seems like it might depend on the method. 8 / 48
Stoquastic Hamiltonians form a distinct complexity class called Stoq-MA. QMA AM Stoq-MA MA NP The decision problem is the ground energy of a stoquastic Hamiltonian. The transverse field Ising model is complete for this class. Stoquastic k -sat ( H = � H a , there exists a | ψ � such that H a | ψ � = 0) is MA-complete 9 / 48
The idea behind our research program is that stoquasticity is basis dependent. So for which Hamiltonians can we find a basis that makes them stoquastic? 10 / 48
Deciding Stoquasticity of 2-Local Hamiltonians Joel Klassen 2018 11 / 48
Hello my name is Joel. I am a postdoc at QuTech Working with Barbara Terhal Thanks for hosting me. 12 / 48
Introduction ◮ This is joint work largely done by myself and Milad Marvian. Our collaborators are Barbara Terhal, Marios Iannous, Itay Hen and Daniel Lidar. ◮ Our research has focused on stoquastic Hamiltonians ◮ In particular I have been trying to develop algorithms for deciding when a Hamiltonian is stoquastic ◮ I will explain what a stoquastic Hamiltonian is ◮ I will give some motivation for why it is interesting ◮ I will present a polynomial time algorithm for deciding if a two-local multiqubit Hamiltonain, with no one-local terms, can be made stoquastic in some local basis. 13 / 48
Yeah but like, what even is a stoquastic Hamiltonian anyways? 14 / 48
Did you say stochastic? ◮ A stochastic process is a random process evolving in time. ◮ A quantum process is not a stochastic process in general. ◮ Indeed unitary time evolution is deterministic. ◮ However some quantum systems can be modelled by stochastic processes. Stochastic + Quantum = Stoquastic 1 1 [Bravyi et.al. (2006)] 15 / 48
We often want to evaluate expressions like this: Tr [ Oe − β H ] One way is to break up the exponentials into small steps, and insert identities. L − 1 � p i | I − β Tr [ Oe − β H ] = � � � p 0 | O | p 1 � LH | p i +1 � p | p 0 = p L i =1 Terms can be evaluated if O and H are local in the basis p i . Morally, this is path integration, with p representing a particular path. � � end | O | beginning �× amplitudes of paths 16 / 48
Boy, it sure would be nice if we didn’t have to evaluate all of those paths... 17 / 48
◮ What if we just sample from these paths according to their weights? Will our answers be faithful? ◮ Not if our amplitudes interfere! Random sampling can obscure important coherence effects. ◮ This is called the sign problem. ◮ Its very much like the difference between burnished metal and a polished mirror. ◮ However if all of our amplitudes are positive and real... then we don’t have this problem. 18 / 48
Enter Stoquastic Hamiltonians Consider a Hamiltonian H such that all of its off diagonal elements are non-positive and real in some basis {| i �} . ◮ For all values of β ≥ 0: � i | e − β H | j � ≥ 0 ◮ Path amplitudes will be positive and real , and we can perform stochastic sampling of our path integrals. ◮ Such a matrix H is an instance of a “Z-matrix” ◮ Matrices of this type are also employed in the study of economics, control theory, and population dynamics. 19 / 48
A Z-matrix in any other basis would smell as sweet. ◮ Critically, a Z-matrix is basis dependent! ◮ Generally one wants to say that when a Hamiltonian can be efficiently transformed into a Z-matrix while preserving sparsity (ie local structure), then it is “stoquastic” (Quantum Stochastic) under that transformation. Stoquastic ≃ Z-matrix in some efficient representation ◮ There is a subtlety here. We can ask that each k -local term be a Z-matrix, or we can ask that the whole Hamiltonian be a Z-matrix ◮ These two questions are distinct! But for two-local qubit Hamiltonians they are the same. 20 / 48
Okay but I mean who cares? 21 / 48
Motivation The Quantum Monte Carlo Community Stoquastic Hamiltonians avoid the sign-problem and thus are more amenable to quantum Monte Carlo methods. Computational Complexity Theorists QMA AM Stoquastic Hamiltonians constitutes a distinct and Stoq-MA interesting computational complexity class: Stoq- MA [Bravyi et.al. (2006)(2008)] [Aharanov, Grillo MA (2019)] NP Adiabatic evolution of frustration free stoquastic Hamiltonians can be simulated efficiently. [Bravyi, Terhal (2008)] 22 / 48
Motivation Experimentalists and Engineers ◮ It seems as though finding ground states and ground energies of stoquastic Hamiltonians is easier than for generic Hamiltonians. ◮ Perhaps in adiabatic quantum computation we want to build devices that are not stoquastic. (eg. TFIM is stoquastic) Theoretical Physicists ◮ Many natural systems are manifestly stoquastic in what is considered a natural basis. (spinless destinguishable particles, hopping bosons) ◮ Is there something deep behind this? 23 / 48
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